Hilbert polynomials of the algebras of $ SL_2 $-invariants

HILBERT POLYNOMIALS OF THE ALGEBRAS OF SL2 -INVARIANTS

arXiv:1102.3290v1 [math.AG] 16 Feb 2011

LEONID BEDRATYUK Abstract. We offer a Maple-procedure for computing of the Hilbert polynomials of the algebras of SL2 -invariants

1. Let K be a field, charK = 0. Let Vd be d+1-dimensional SL2 -module of binary forms of degree d and let Vd = Vd1 ⊕ Vd2 ⊕ · · · ⊕ Vdn , d = (d1 , d2, . . . , ds ). Denote by K[Vd ]SL2 the algebra of polynomial SL2 -invariant functions on Vd . It is well-known that the algebra Id := K[Vd ]SL2 is finitely generated and graded : Id = (Id )0 ⊕ (Id )1 ⊕ (Id )2 ⊕ · · · ⊕ (Id )n ⊕ + · · · , where (Id )n is a vector K-space of invariants of degree n. The Hilbert function of the algebra Id is defined as dimension of the vector space (Id )n : H(Id , n) = dim(Id )n . It is well-known [1]-[3] that the Hilbert function of finitely generated graded K-algebra is equal (starting from some n) to a polynomial of n : H(Id , n) = h0 (n)nr + h1 (n)nr−1 + · · · , where hi (n) is some periodic function with values in Q. Then such a polynomial is called the Hilbert polynomial of graded algebra. From combinatorial point of view the Hilbert polynomials are so-called quasi-polynomials, see [4], Chapter 4. In this short notes we present a Maple-procedure for calculation of the Hilbert polynomials for the algebra Id . 2. For the case of one binary form (s = 1) there exists ([5], [6]) classical Cayley-Sylvester formula for calculation of values of Hilbert function of Id : H(Id , n) = ωd (n, 0) − ωd (n, 2) , where ωd (n, k) is the number non-negative integer solutions of the system: ( dn−k α1 + 2α2 + · · · + d αd = , 2 α0 + α1 + · · · + αd = n. For generalisation of the Cayley-Sylvester formula to Id see [8]. Also, see [5], [6], we have:  ! h nd i d (1 − q) , H(Id , n) = q 2 n q   d here is q-binomial coefficient: n q   (1 − q d+1 )(1 − q d+2 ) . . . (1 − q d+n ) d , := n q (1 − q)(1 − q 2 ) . . . (1 − q n ) 1

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h nd i nd and q 2 denotes the coefficient of q 2 . But it seems impossible to extract the Hilbert polynomials from this representation of the Hilbert function. To calculate the Hilbert polynomials we have used our knowledge of Poincare series of the algebras of invariants Id , see [7]. The following statement is our main computational tool : Theorem. Let the Poincar´e series of a finitely generated graded K-algebra A has the form R(z) P(A, z) = , Q(z) the polynomials R(z), Q(z), deg R(z) < deg Q(z) are coprime and let λ1 , λ2 , . . . λr are the roots of the denominator Q(z) with the multiplicities k1 , k2 , . . . , kr . Then the Hilbert polynomials of A has the form H(A, n) = τk1 (n)λk11 + τk2 (n)λk12 + · · · + τkr (n)λk1r , n = 0, 1, 2, . . . ,

here τki (n) is a polynomial of n with rational coefficients of degree ≤ ki − 1. The proof follows from [4] Theorem 4.1.1 and Proposition 4.1.1 Example. Let us calculate the Hilbert polynomial for the algebra of joint invariants of two binary forms of degrees 2 and 3. The Poincare series equals: z6 − z5 + z4 − z3 + z2 − z + 1 = P(I2,3 , z) = (1 − z 3 ) (1 − z 5 ) (1 − z + z 2 − z 3 ) (1 − z 2 ) = 1 + z 2 + z 3 + 2 z 4 + 2 z 5 + 3 z 6 + 4 z 7 + 5 z 8 + 6 z 9 + 8 z 10 + 9 z 11 + 12 z 12 + · · ·

The denominator



Q(z) = 1 − z 3 1 − z 5 1 − z + z 2 − z 3 1 − z 2 = z 13 − z 12 − z 10 + z 7 + z 6 − z 3 − z + 1,

has the root 1 with the multiplicity 4 (the transendence degree of I2,3 ) and the 9 roots q q √ √ 1 1√ 1 1 1√ 1 −1, i, −i, − + 5 + i 10 + 2 5, − − 5 + i 10 − 2 5, 4 4 q4 4 q4 4 √ √ 1 1√ 1 1 1√ 1 1 1 √ 1 1 √ − − 5 − i 10 − 2 5, − + 5 − i 10 + 2 5, − + i 3, − − i 3, 4 4 4 4 4 4 2 2 2 2 each of multiplicity 1. Thus, we are looking the Hilbert polynomial in the following form ! √ q √ 1 1 5 − i 10 − 2 5 + H(I2,3 , n) = C0 + C1 n + C2 n2 + C3 n3 + C4 + C5 − − 4 4 4     q q √ √ 1 1√ 1 1√ 1 1 5 + i 10 − 2 5 + C7 − + 5 − i 10 + 2 5 − iC8 + +C6 − − 4 4 4 4 4 4 ! √     q √ 1 1 √ 1 5 1 1 √ 1 +C9 − + i 3 + C10 − − i 3 + iC11 + C12 − + + i 10 + 2 5 . 2 2 2 2 4 4 4 Taking into account the initial conditions H(I2,3 , 0) = 1, H(I2,3 , 1) = 0, H(I2,3 , 2) = 1, . . . H(I2,3 , 12) = 12 we solve the linear system of equations for Ci and after simplification get the Hilbert polynomial:     1 3 2 7 2 1 1 2 1 √ H(I2,3 , n) = 5 cos n + n + n+ cos πn − πn + 360 240 6 10 5 50 5

HILBERT POLYNOMIALS

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  √       q √ 1 5 1 1 4πn 4πn 1πn 2πn + + + + cos − cos cos 50 + 10 5 sin 50 q 10  5 50  5  2   5  8 √ 4 1 1 1 √ 1 2 1 2 10 − 2 5 sin 3 sin − π n + cos πn + π n − sin πn + 25 5 9 3 27 3 8 2     q q √ √ 1 1 √ 497 2 4 7 cos (π n) − πn − πn + . 10 + 2 5 sin 5 10 − 2 5 sin + 32 25 5 50 5 1440 3. To calculate the Hilbert polynomial of the SL2 -module Vd , d = (d1 , d2 , . . . , dn ) we offer the procedure HilbertPol that included in our updated Maple-package Poincare_Series, [7]: Command name: HilbertPol Feature: Computes the Hilbert polynomial for the algebras of joint invariants for the binary forms of degrees d1 , d2 , . . . , dn . Calling sequence: HilbertPol([d1 , d2 , . . . , dn ]) ; Parameters: [d1 , d2, . . . , dn ] - a list of degrees of n binary forms. n - an integer, n ≥ 1. Below are listed of results of some calculations. 3.1 H(I2 , n) > dd:=[2]:HilbertPol(dd); 1 1 cos (π n) + 2 2 3.2 H(I3 , n) > dd:=[3]:HilbertPol(dd); 1 1 cos (π n) + cos 4 2



 1 1 πn + 2 4

3.3 H(I4 , n) > dd:=[4]:HilbertPol(dd);     2 1 1 1 1√ 5 2 3 sin n + cos (π n) + cos πn − πn + 6 4 3 3 9 3 12 3.4 H(I5 , n) > dd:=[5]:HilbertPol(dd); 

     1 3 1 1 1 107 1 2 n + n+ cos (π n) + cos (π n) + cos πn + cos (π n) + 384 384 64 64 32 2 576         1 1 1 1 1 2 1 1 + cos π n + cos πn + sin πn + cos πn − 9 3 9 3 16 4 16 4       1 3 1 3 9 1 107 − sin πn + cos πn + cos πn + 16 4 16 4 32 2 576

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3.5 H(I6 , n) > dd:=[6]:HilbertPol(dd);

    7 7 37 1 1 2 3 n + n+ n + cos (π n) + cos (π n) + 1440 64 960 64 320       q q √ √ 1 √ 2 2 1 1 √ 4 + 5 10 + 2 5 sin 10 + 2 5 sin 5 cos πn − πn − πn + 100 5 100 5 50 5       q q √ √ 4 1 √ 1 4 1 4 cos πn − πn − πn + 5 10 − 2 5 sin 10 − 2 5 sin + 10 5 100 5 100 5         1 2 1 √ 1 1 1 1 9 2 + cos πn + πn − sin πn + cos πn + cos (π n) + 3 sin 9 3 27 3 16 2 16 2 32     2 1 1 √ 497 2 5 cos + cos πn + πn + 10 5 50 5 1440

3.6 H(I(1,1) , n) > dd:=[1,1]:HilbertPol(dd); 1 1 cos (π n) + 2 2

3.7 H(I(1,2) , n) > dd:=[1,2]:HilbertPol(dd);     1 1 1√ 5 2 2 1 n + cos (π n) + cos πn − πn + 3 sin 6 4 3 3 9 3 12 3.8 H(I(1,3) , n) > dd:=[1,3]:HilbertPol(dd);



     3 1 3 3 1 1 2 n + n+ cos (π n) + cos πn + cos (π n) + 64 64 16 2 32 32   7 9 1 7 + cos (π n) + cos πn + 32 16 2 32

3.9 H(I(1,4) , n) > dd:=[1,4]:HilbertPol(dd);

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      1 3 7 2 1 √ 79 1 2 2 3 sin n+ n + n + cos πn − πn + 540 360 27 3 27 3 540       q √ 1 √ 1 4 2 4 1 √ πn − πn + cos πn − 5 10 + 2 5 sin 5 cos + 100 5 50 5 10 5       q q √ √ 1 √ 7 2 1 4 4 πn − πn + cos πn − 10 − 2 5 sin 5 10 − 2 5 sin − 100 5 100 5 27 3       7 √ 3 1 2 1 √ 2 2 − πn + cos (π n) + cos πn + πn − 3 sin 5 cos 81 3 16 10 5 50 5   q √ 1 2 763 10 + 2 5 sin − πn + 100 5 2160 3.10 H(I(1,2,3) , n) > dd:=[1,2,3]:HilbertPol(dd);   17 17 4 1033 3 689 1 5 n2 + n + n + n + cos (π n) + 172800 7680 51840 512 7680         4 2 1 1 46667 3 3 9 + n− cos πn − sin πn + cos πn + cos (π n) + 81 3 64 2 64 2 512 207360       q √ 2 3 1 2 1 √ 2 πn + cos πn − πn + − 5 cos 10 + 2 5 sin 50 5 50 5 100 5       q √ 3 4 1 √ 2 4 1 √ πn + cos πn + πn − 5 10 + 2 5 sin 5 cos + 500 5 50 5 50 5       q q √ √ 1 √ 4 4 1 2 2 − 5 10 − 2 5 sin 10 − 2 5 sin πn − π n + cos πn + 500 5 100 5 9 3     4 √ 1 15 1 141 3 3 sin (2/3 π n) − + sin πn + cos πn + cos (π n) + 243 16 2 64 2 1024 65827 + 230400 The package can be downloaded from the web: sites.google.com/site/bedratyuklp/. References [1] R. Stanley, Hilbert functions of graded algebras, Adv. Math. 28, 57-83 (1978). [2] L. Robbiano, Introduction to the Theory of Hilbert Function. Queen’s Papers in Pure and Applied Mathematics 85 (1990),1–26. [3] D. Eisenbud, The geometry of syzygies. A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics 229. New York, NY: Springer, 243 p. (2005) [4] R. Stanley, Enumerative combinatorics. Vol. 1., Cambridge Studies in Advanced Mathematics. 49. Cambridge: Cambridge University Press. (1999). [5] D. Hilbert, Theory of algebraic invariants, Cambridge University Press, 1993. [6] T. Springer, Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag,1977. [7] L.Bedratyuk, The MAPLE package for calculating Poincar´e series, arXiv:1006.5372 [8] L. Bedratyuk, Weitzenb¨ock derivations and the classical invariant theory, I: Poincar´e series. Serdica Math. J., vol. 36, No 2, 2010, 99-120.